On some hypersurfaces of high-dimensional tori related with the Riemann zeta-function
Let R be a rectangle in the complex plane, with sides parallel to the real and imaginary axes, and let \(L(T)=mes\{t\in (0,T): \log\zeta(\sigma+it)\in R\}.\) Then it was shown over 50 years ago by Bohr and Jessen that \(L(T)/T\) tends to a limit \(W(\sigma)\) for any \(\sigma >1/2\) which is not the real part of a zero. The first author [Acta Arith. 48, 167-190 (1987; Zbl 0569.10019)] has refined this to obtain \[ L(T)=W(\sigma)T+O(T(\log \log T)^{-(\sigma -1)/7+\epsilon}) \] for \(\sigma >1\), and a similar result [Acta Arith. (to appear)] for \(1/2<\sigma \leq 1\). These results depend on a technical estimate which has been postponed to the present paper. Consider the N-dimensional volume \(V_ N(R,\epsilon)\) of points of distance at most \(\epsilon\) from the boundary of the set \(\{(x_ 1,...,x_ N)\in [0,1)^ N:S(x_ 1,...,x_ N)\in R\}.\) Here \(S(x_ 1,...,x_ N)=-\sum^{N}_{n=1}\log (1-p_ n^{-\sigma} \exp (2\pi ix_ n)).\) One then has \[ V_ N(R,\epsilon)\ll a^{-1}\epsilon N+(aN)^{2N\sigma},\text{ for any } a>0. \]
scientific article; zbMATH DE number 4040013